Sheath-Plasma Criterion of Fluid Theory with Finite Ion Temperature

with a polytropic index . The value of is important in relation to probe measurements, and one may set 1⁄4 3 for an adiabatic case (one-dimension), or 1⁄4 1 for an isothermal case. In fact, varies from three to unity owing to limited heat flow, since there is no heat flow in the adiabatic case and unhindered heat flow in the isothermal case. The value of for the sheath-plasma criterion has remained a problem. To derive the sheath-plasma criterion, equations applicable in a sheath region with space charge are used in general. However, there is another method of deriving the sheath-plasma criterion, in which equations applicable in a plasma or presheath region are used. Since quasi-neutrality is assumed in the equations in the second method, the equations become invalid in the sheath region owing to space charge, and a singularity gives a sheath-plasma boundary, where the ion drift speed is given by eq. (1) for Ti 1⁄4 0. The extension of the second method for Ti 61⁄4 0 is achieved by considering ion pressure, and shows the sheathplasma criterion indicating that the ion drift speed at the sheath-plasma boundary is given by eq. (2). As for the first method for Ti 61⁄4 0, the generalized sheath-plasma criterion of the ion velocity distribution function is obtained, although the details of the ion velocity distribution function are not determined. The purpose of this report is to find a solution of the equations in the second method for a finite ion temperature, which may vary in accordance with the polytropic approximation, because only the solution under the condition of Ti 1⁄4 0 or Ti 1⁄4 const: is known. Basic equations of the fluid theory treating a steady-state one-dimensional plasma with a uniform ion generation G (1⁄4 const:) are written as